5.3 Summary and conclusions
6.1.1 Drying at reference temperature
Assuming that the changes in the microstructure of the concrete due to drying can be neglected, the most important factor to determine the type of transport mechanism dom- inating the water transport process is the moisture content (see Chapter 2.2.1). There- fore, in a simplified diffusion approach to calculate water transport in concrete, the dif- fusion coefficient has to be define as dependent on the moisture content or rather on the relative humidity of the concrete pores. The approach from Bažant and Najjar [11]
defines the influence of the relative humidity in the concrete pores on the diffusion coef- ficient by means of the relative humidityh(see Eq. 2.6 in Chapter 2.2.3). Following this approach, the parameters of the equation f(h)can be calibrated to fit the measurements.
With the parameters D1 = 15 mm2/d, α0= 0.02,n= 8, and hc = 0.83 a good confor- mity between model and measurements can be obtained for the concrete MRC drying at 20 °C and 65 % RH (see continuous line in Fig. 6.1). However, if the concrete is subject to drying at a different environmental relative humidity, like for instance, 85 %, with the parameters calculated for 65 % RH the model does not follow the measurements accurately (see dashed line in Fig. 6.1).
According to the model from Bažant and Najjar, the diffusion coefficient is the higher, the higher the relative humidity of the concrete pores. As seen in Fig. 6.1, this approach implies that samples drying at higher environmental relative humidities reach the equi- librium with the environment faster than those drying at lower relative humidities. As it was discussed in Chapter 4.1.1.1 that could not be verified by the experiments. The experiments showed that the mean values of the relative humidity for concretes drying at 85 % and 65 % RH tend to reach the equilibrium with the environment at a similar point in time, as it would be the case if the diffusion coefficient were constant. The generality of the model as it was presented by Bažant and Najjar is very poor because it requires a new calibration for every time the relative humidity of the environment changes.
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 6 5
7 0 7 5 8 0 8 5 9 0 9 5 1 0 0
m o d e l c a l i b r a t e d t o f i t t h e m e a s u r e m e n t s Mean relative humidity, hmean [%]
D u r a t i o n o f d r y i n g , t [ d ]
R H m e a s u r e d c a l c u l a t e d 8 5 %
6 5 %
p r e d i c t i o n f r o m c a l i b r a t e d m o d e l
Figure 6.1: Comparisons between the measured values of concrete MRC drying at 20 °C and 85 % as well as 65 % RH and the calculated values using the model from Bažant and Najjar [11]
The lack of generality of the model can be overcome by including in the function that accounts for the influence of the moisture content on the diffusion coefficient of concrete two additional dependencies, namely the relative humidity of the environment and the gradient of the relative humidity. Hence, the function f(h)presented in Chapter 2.2.3 becomes f(h,h∞,dh/dr). After trying different configurations, the following definition was selected because of its simplicity and the good accuracy that the model provides when compared to the experimental measurements.
f(h,h∞,dh
dr) = 1
1+
1−h 1−hc(h∞,dh/dr)
n (6.3)
hc(h∞,dh/dr) =0.5+ 0.5
1+2·(1−h∞)2−0.5·
dh dr
·r0
0.5
(6.4) In the formulation presented abovehcis not a constant anymore but rather a function of the relative humidity of the environmenth∞and the absolute gradient of the relative hu- midity|dh/dr|. This implies that the relative humidity at which the function drops down, ruled by the value ofhc, is dependent on the combined effects of the environmental rel- ative humidity and the gradient of the relative humidity and moves towards lower values as the relative humidity in the concrete decreases. Furthermore, being the parameter hc variable, the parameter α0, as presented in Eq. 2.7 from Chapter 2.2.3, is not re- quired and can be taken as zero which implies that the function f(h,h∞,dh/dr)decreases
6.1 Modelling the transport of moisture in concrete
continuously as the relative humidity of the pores approaches 0 %. Eq. 6.4 includes the constant r0which equals 1 mm. In case the model is run in different dimensions than mm, this constant has to be converted to the correspondent units of length.
In case of drying at 20 °C, the model has two material dependent parameters, namelyn (see Eq. 6.3) andD1(see Eq. 2.5 in Chapter 2.2.3). This model was compared with the measurements of relative humidity in the pore cavities conducted at 20 °C for all three concrete mixtures drying at 85 % and 65 % RH. Based on these comparisons the para- metersnandD1were calibrated. Table 6.1 presents the calibrated parameters according to the w/c-ratio of the concrete samples.
Table 6.1: Parameters of the model calibrated according to the experimental results
w/c-ratio D1 n
[-] [mm²/d] [-]
MLC 0.40 5.4 6.4
MRC 0.50 20.0 5.7
MHC 0.60 22.0 4.9
Concrete Mixture
The generality of the model can be tested if after calibrating its parameters with the measured data, they can be approximated through defined functions. These functions have to be based on the characteristics of the concrete microstructure best described by the w/c-ratio. Eqs. 6.5 and 6.6 present the dependency of the parametersD1andnon the w/c-ratio of the concrete mix based on the calibrations conducted with the experimental results from the concretes MLC (w/c = 0.4), MRC (w/c = 0.5) and MHC (w/c = 0.6).
D1=2.2+ 19.8
1+ (1.83·(1−w/c))25 (6.5) n=6.1− 1.8
1+ (2.3·(1−w/c))9 (6.6) The values of the diffusion coefficient at reference conditionsD1are given in mm2/d.
According to Eq. 6.5, for concretes without capillary porosity (i.e. w/c-ratio < 0.4),D1
approaches a constant value around 2.2 mm2/d, while by concretes containing capillary porosity, D1 increases steeply towards reaching a maximum value of 22 mm2/d. The exponentncontrols how steep the diffusion coefficient decreases with decreasing pore relative humidity and is the higher the lower the w/c-ratio of the concrete. In Eq. 6.6,nis described by a s-shaped function with a maximum value of 6.1, for w/c-ratios lower than 0.4, and a minimum value of around 4.3, for w/c-ratios higher than 0.7. The equations that compose the model of moisture transport are summarized in Appendix E.1.
In Fig. 6.2 the mean values of pore relative humidity calculated from the measurements conducted on the concretes MLC and MHC are compared with the calculations from the model using the parameters from Table 6.1.
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0
6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0
6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0
Mean relative humidity, hmean [%]
D u r a t i o n o f d r y i n g , t [ d ] R H m e a s u r e d c a l c u l a t e d 8 5 % 6 5 % C o n c r e t e M L C d r y i n g a t 2 0 ° C
D u r a t i o n o f d r y i n g , t [ d ] R H m e a s u r e d c a l c u l a t e d 8 5 % 6 5 % C o n c r e t e M H C d r y i n g a t 2 0 ° C
Figure 6.2: Comparisons between the calculated values using the model of moisture transport and the measured values of relative humidity from the concretes MLC (left) and MHC (right) drying at 20 °C and 85 % as well as 65 % RH
A very good concordance between model calculations and measurements is achieved for both concretes drying at 85 % and 65 % RH. Similar results were seen by the concrete MRC presented in Appendix D.1.
6.1.1.1 Comparison of the model calculation with experimental results from the literature
Very few measurements of pore relative humidity have been conducted and published in the literature, therefore a data base to further improve the calibration of the model and validate their results is not available. One of the few well documented experiments was conducted by Hanson [73] in the late sixties of the last century. Hanson recorded the development of the pore relative humidity of a concrete cylinder with a diameter of 150 mm drying at 20 °C and 50 % RH during three years. The w/c-ratio of the concrete mixture investigated by Hanson was 0.657. He used four Monfore relative humidity probes [99] and placed them in different positions over the radius to assess the humidity profile of the sample. The comparison of the model with the results from Hanson is presented in Fig. 6.3. The model can reproduce the behaviour of the concrete sample measured by Hanson very well even providing that the environmental relative humidity, w/c-ratio, and member size go beyond the ranges considered in the present investigation.
6.1 Modelling the transport of moisture in concrete
0 1 5 0 3 0 0 4 5 0 6 0 0 7 5 0 9 0 0 1 0 5 0 1 2 0 0
5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0
Mean relative humidity, hmean [%]
D u r a t i o n o f d r y i n g , t [ d ]
R H m e a s u r e d c a l c u l a t e d 5 0 %
Figure 6.3: Comparisons between the calculated values using the model of water transport and the meas- ured values of mean relative humidity from a concrete drying at 20 °C and 50 % RH according to Hanson [73]
Nevertheless, assuming that the function that accounts for the influence of the moisture content on the diffusion coefficient of concrete f(h,h∞,dh/dr)is dependent on the rel- ative humidity of the environmenth∞, implies that the diffusion coefficient which is an intrinsic property of the material is influenced by a condition that takes place outside the concrete microstructure. This is on first sight not quite correct, however, the en- vironmental relative humidity denotes the final value of humidity to be reached in the concrete pores and under this consideration it is included in the calculation of the dif- fusion coefficient. Another aspect that needs to be discussed is the size of the concrete member. The dependency of the diffusion coefficient on the relative humidity of the en- vironment as defined by Eq. 6.3 may vary considerably if the geometry of the concrete members to evaluate differs from the one used for the calibration. The diameter of the sample used by Hanson doubles the diameter of the samples used in the present inves- tigation. However, even when the model shows very similar results when compared with the measurements from Hanson, a conclusive statement about the validity of the model for massive concrete members cannot be made.
6.1.1.2 Modelling concrete wetting
Although the measurements of concrete internal relative humidity were conducted only for drying specimens, the wetting of the concrete can also be calculated with the model.
Under the assumption that the process of drying and wetting follow the same time de- pendency, Eqs. 6.3 and 6.4 must be changed as follows.
f(h,h∞,h0,dh
dr) = 1
1+
1−h∞+h−h0
1−hc(h0,dh/dr)
n (6.7)
hc(h0,dh/dr) =0.5+ 0.5
1+2·(1−h0)2−0.5·
dh dr
·r0
0.5
(6.8) Eq. 6.7 corresponds to the mirror image, with respect to a horizontal line cutting the function in two parts, of Eq. 6.3 which was formulated for drying. In order to model the process of wetting, the relative humidity of the environmenth∞must be replaced by the initial relative humidity of the concrete poresh0for the definition ofhc(h0,dh/dr). Ad- ditionally, in the definition of f(h,h∞,h0,dh/dr)the initial and end value of the relative humidity,h0andh∞, need to be considered.